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Consistency algorithms

Consistency algorithms. Chapter 3. Consistency methods. Approximation of inference: Arc, path and i-consistecy Methods that transform the original network into a tighter and tighter representations. Arc-consistency. X. Y. . 1,. 2,. 3. 1,. 2,. 3. 1  X, Y, Z, T  3 X  Y Y = Z

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Consistency algorithms

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  1. Consistency algorithms Chapter 3

  2. Consistency methods • Approximation of inference: • Arc, path and i-consistecy • Methods that transform the original network into a tighter and tighter representations ICS 275A - Constraint Networks

  3. Arc-consistency X Y  1, 2, 3 1, 2, 3 1  X, Y, Z, T  3 X  Y Y = Z T  Z X  T  = 1, 2, 3 1, 2, 3  T Z ICS 275A - Constraint Networks

  4. 1 3 2 3 Arc-consistency X Y  1  X, Y, Z, T  3 X  Y Y = Z T  Z X  T  =  T Z ICS 275A - Constraint Networks

  5. Arc-consistency ICS 275A - Constraint Networks

  6. Revise for arc-consistency ICS 275A - Constraint Networks

  7. Figure 3.3: (a) Matching diagram describing a network of constraints that is not arc-consistent (b) An arc-consistent equivalent network. ICS 275A - Constraint Networks

  8. AC-1 • Complexity (Mackworth and Freuder, 1986): • e = number of arcs, n variables,k values • (ek^2, each loop, nk number of loops), best-case = ek, • Arc-consistency is: ICS 275A - Constraint Networks

  9. AC-3 • Complexity: • Best case O(ek), since each arc may be processed in O(2k) ICS 275A - Constraint Networks

  10. Example: A three variable network, with two constraints: z divides x and z divides y (a) before and (b) after AC-3 is applied. ICS 275A - Constraint Networks

  11. AC-4 • Complexity: • (Counter is the number of supports to ai in xi from xj. S_(xi,ai) is the set of pairs that (xi,ai) supports) ICS 275A - Constraint Networks

  12. Example applying AC-4 ICS 275A - Constraint Networks

  13. Distributed arc-consistency(Constraint propagation) • Implement AC-1 distributedly. • Node x_j sends the message to node x_i • Node x_i updates its domain: • Messages can be sent asynchronously or scheduled in a topological order ICS 275A - Constraint Networks

  14. Exercise: make the following network arc-consistent • Draw the network’s primal and dual constraint graph • Network = • Domains {1,2,3,4} • Constraints: y < x, z < y, t < z, f<t, x<=t+1, Y<f+2 ICS 275A - Constraint Networks

  15. Arc-consistency Algorithms • AC-1: brute-force, distributed • AC-3, queue-based • AC-4, context-based, optimal • AC-5,6,7,…. Good in special cases • Important:applied at every node of search • (n number of variables, e=#constraints, k=domain size) • Mackworth and Freuder (1977,1983), Mohr and Anderson, (1985)… ICS 275A - Constraint Networks

  16. Is arc-consistency enough? • Example: a triangle graph-coloring with 2 values. • Is it arc-consistent? • Is it consistent? • It is not path, or 3-consistent. ICS 275A - Constraint Networks

  17. Path-consistency ICS 275A - Constraint Networks

  18. Path-consistency ICS 275A - Constraint Networks

  19. Revise-3 • Complexity: O(k^3) • Best-case: O(t) • Worst-case O(tk) ICS 275A - Constraint Networks

  20. PC-1 • Complexity: • O(n^3) triplets, each take O(k^3) steps  O(n^3 k^3) • Max number of loops: O(n^2 k^2) . ICS 275A - Constraint Networks

  21. PC-2 • Complexity: • Optimal PC-4: • (each pair deleted may add: 2n-1 triplets, number of pairs: O(n^2 k^2)  size of Q is O(n^3 k^2), processing is O(k^3)) ICS 275A - Constraint Networks

  22. Example: before and after path-consistency • PC-1 requires 2 processings of each arc while PC-2 may not • Can we do path-consistency distributedly? ICS 275A - Constraint Networks

  23. Path-consistency Algorithms • Apply Revise-3 (O(k^3)) until no change • Path-consistency (3-consistency) adds binary constraints. • PC-1: • PC-2: • PC-4 optimal: ICS 275A - Constraint Networks

  24. I-consistency ICS 275A - Constraint Networks

  25. Higher levels of consistency, global-consistency ICS 275A - Constraint Networks

  26. Revise-i • Complexity: for binary constraints • For arbitrary constraints: ICS 275A - Constraint Networks

  27. 4-queen example ICS 275A - Constraint Networks

  28. I-consistency ICS 275A - Constraint Networks

  29. Arc-consistency for non-binary constraints:Generalized arc-consistency Complexity: O(t k), t bounds number of tuples. Relational arc-consistency: ICS 275A - Constraint Networks

  30. Examples of generalized arc-consistency • x+y+z <= 15 and z >= 13 implies x<=2, y<=2 • Example of relational arc-consistency ICS 275A - Constraint Networks

  31. More arc-based consistency • Global constraints: e.g., all-different constraints • Special semantic constraints that appears often in practice and a specialized constraint propagation. Used in constraint programming. • Bounds-consistency: pruning the boundaries of domains ICS 275A - Constraint Networks

  32. Example for alldiff • A = {3,4,5,6} • B = {3,4} • C= {2,3,4,5} • D= {2,3,4} • E = {3,4} • Alldiff (A,B,C,D,E} • Arc-consistency does nothing • Apply GAC to sol(A,B,C,D,E)? •  A = {6}, F = {1}…. • Alg: bipartite matching kn^1.5 • (Lopez-Ortiz, et. Al, IJCAI-03 pp 245 (A fast and simple algorithm for bounds consistency of alldifferent constraint) ICS 275A - Constraint Networks

  33. Global constraints • Alldifferent • Sum constraint (variable equal the sum of others) • Global cardinality constraint (a value can be assigned a bounded number of times to a set of variables) • The cummulative constraint (related to scheduling tasks) ICS 275A - Constraint Networks

  34. Bounds consistency ICS 275A - Constraint Networks

  35. Bounds consistency for Alldifferent constraints ICS 275A - Constraint Networks

  36. Boolean constraint propagation • (A V ~B) and (B) • B is arc-consistent relative to A but not vice-versa • Arc-consistency by resolution: res((A V ~B),B) = A Given also (B V C), path-consistency: Res((A V ~B),(B V C) = (A V C) What will generalized arc-consistency can do to cnfs? Relational arc-consistency rule = unit-resolution ICS 275A - Constraint Networks

  37. Boolean constraint propagation Example: party problem • If Alex goes, then Becky goes: • If Chris goes, then Alex goes: • Query: Is it possible that Chris goes to the party but Becky does not? ICS 275A - Constraint Networks

  38. Gausian and Boolean propagation • Linear inequalities • Boolean constraint propagation ICS 275A - Constraint Networks

  39. Constraint propagation for Boolean constraints: Unit propagation ICS 275A - Constraint Networks

  40. Consistency for numeric constraints ICS 275A - Constraint Networks

  41. Tractable classes ICS 275A - Constraint Networks

  42. Changes in the network graph as a result of arc-consistency, path-consistency and 4-consistency. ICS 275A - Constraint Networks

  43. Distributed arc-consistency(Constraint propagation) • Implement AC-1 distributedly. • Node x_j sends the message to node x_i • Node x_i updates its domain: • Generalized arc-consistency can be implemented distributedly: sending messages between constraints over the dual graph: ICS 275A - Constraint Networks

  44. Distributed Arc-Consistency • Arc-consistency can be formulated as a distributed algorithm: A B C D F G a Constraint network ICS 275A - Constraint Networks

  45. Relational Arc-consistency A The message that R2 sends to R1 is R1 updates its relation and domains and sends messages to neighbors B C D F G ICS 275A - Constraint Networks

  46. 1 A A 3 A 2 AB AC B A C B 5 4 ABD BCF 6 F D DFG DRAC on the dual join-graph ICS 275A - Constraint Networks

  47. Distributed Relational Arc-Consistency • DRAC can be applied to the dual problem of any constraint network: ICS 275A - Constraint Networks

  48. A A A 1 1 1 2 3 2 3 3 B A A B A 1 2 1 1 1 1 3 2 3 3 C A A A 2 1 3 2 1 1 1 2 3 2 2 2 3 1 3 3 3 3 2 C B F B D B A A B A 2 1 1 1 1 2 1 1 1 1 2 2 1 3 2 3 3 3 2 2 1 3 3 3 2 3 3 1 3 2 F D 1 1 2 3 3 Iteration 1 Node 2 sends messages Node 1 sends messages Node 3 sends messages Node 4 sends messages Node 5 sends messages Node 6 sends messages 1 A A 3 A 2 AB AC A A A C AB B 5 4 ABD BCF B 6 F D DFG ICS 275A - Constraint Networks

  49. 1 A A 3 A 2 AB AC B A C B 5 4 ABD BCF 6 F D DFG Iteration 1 ICS 275A - Constraint Networks

  50. B 1 1 A 3 A 3 A 2 AB AC B A C B 5 4 ABD BCF 6 F D DFG Iteration 2 ICS 275A - Constraint Networks

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